Abstract
Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebrasW B1 + ∞ andw B1 + ∞ of theW-infinity algebrasW 1 + ∞ andw 1 + ∞ are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantumW-infinity algebraW B1 + ∞ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, theseW B1 + ∞ symmetries are shown to be contracted intow B1 + ∞ symmetries of the dispersionless BKP hierarchy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lebedev, D. and Manin, Yu., Conservation laws and Lax representation on Benny's long wave equations,Phys. Lett. 74A, 154–156, (1979); Kodama, Y., A method for solving the dispersionless KP equation and its exact solutions,Phys. Lett. 129A, 223-226 (1988); Solutions of the dispersionless Toda equation,Phys. Lett. 147A, 477-482: (1990); Kodama, Y. and Gibbons, J., A method for solving the dispersionless MP hierarchy and its exact solutions, II,Phys. Lett. 135A, 167-170, (1980).
Dijkgraaf, R., Verlinde, H., and Verlinde, E., Topological strings ind < 1,Nuclear Phys. B352, 59–86, (1991); Krichever, I. M., The dispersionless Lax equations and topological minimal models,Comm. Math. Phys. 143, 415-426 (1991); Dubrovin, B. A., Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models,Comm. Math. Phys. 145, 195-207, (1992).
Bakas, I., The structure of theW ∞ algebra,Comm. Math. Phys. 134, 487–508, (1990); Park, Q-Han, Extended conformal symmetries in read heavens,Phys. Lett. 236B, 429-432, (1990); Takasaki, K. and Takebe, T., SDiff(2) Toda equation - hierarchy, tau function and symmetries,Lett. Math. Phys. 23, 205-214 (1991); SDiff(2) KP hierarchy, inInfinite Analysis, RIMS Research Project 1991,Int. J. Mod. Phys. A7, Suppl. 1B, 889-922 (1992).
Takasaki, K. and Takebe, T., Quasi-classical limit of KP hierarchy,W-symmetries and free fermions, Kyoto preprint KUCP-0050/92, July, 1992; Quasi-classical limit of Toda hierarchy anW-infinity symmetries,Lett. Math. Phys. 28, 165–176 (1993).
Date, E., Kashiwara, M., Jimbo, M., and Miwa, T., Transformation groups for soliton equations, in M. Jimbo and T. Miwa (eds),Nonlinear Integrable Systems - Classical Theory and Quantum Theory, World Scientific, Singapore, 1983.
Orlov, A. Yu. and Schulman, E. I., Additional symmetries for integrable equations and conformal algebra representation,Lett. Math. Phys. 12, 171–179 (1986); Orlov, A. Yu., Vertex operators,∂-problems, symmetries, variational indentities and Hamiltonian formalism for 2 + 1 integrable systems, in V. G. Bar'yankhtaret al. (eds),Plasma Theory and Nonlinear and Turbulent Processes in Physics, World Scientific, Singapore, 1988; Grinevich, P. G. and Orlov, A. Yu., Virasoro action on Riemann surfaces, Grassmannians, det∂ j and Segal Wilsonτ function, in A. A. Belavinet al. (eds),Problems of Modern Quantum Field Theory, Springer-Verlag, New York, 1989.